#7631040

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(6,11)(7,12)(13,14)(15,24)(16,18)(17,22)(19,21)(20,25)(23,27)(26,28);(5,6,10,11)(7,8,12,9)(13,15)(14,20)(16,26)(17,23)(18,22)(19,25)(21,24)(27,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(14,21)(15,25)(16,23)(22,28);(5,8,10,9)(6,12,11,7)(13,16,19,23)(14,18,21,27)(15,26,25,17)(20,22,24,28);(5,9,10,8)(6,7,11,12)(13,17,19,26)(14,22,21,28)(15,16,25,23)(18,20,27,24);(13,18)(14,23)(15,22)(16,21)(17,20)(19,27)(24,26)(25,28)

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313631
maximal normal subgroup of	#72313631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538236);